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The wheels on the bus go.. (Posted on 2003-06-23) Difficulty: 4 of 5

If you drew a dot on the edge of a wheel and traced the path of the dot as the wheel rolled one complete revolution along a line, then the path formed would be called a cycloid (shown below), combining both forward and circular motion.

If a wheel of radius 1 traces out such a path, what is the length of the path formed by one complete revolution?

See The Solution Submitted by DJ    
Rating: 4.3571 (14 votes)

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Solution re: Solution | Comment 3 of 10 |
(In reply to Solution by friedlinguini)

Aha! Half-angle identity to the rescue!

1 - cos t = 2 sin² (t/2)

The formula for the length, then, is Integral(s(t)) evaluated from 0 to 2π.

Integral(s(t)) = Integral(√(2 - 2cos t))
= Integral(√4sin² t/2))
= Integral(|2sin t/2|)

Note that we are only interested in values of t between 0 and 2π. For these values, 2sin t/2 is always positive. This lets us drop the absolute value part

= Integral(2sin t/2)
= -4cos t/2

Evaluating from 0 to 2π gives

-4cos π + 4cos 0 = 8.
  Posted by friedlinguini on 2003-06-23 06:30:43

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